Evaluation of EGM08 based on GPS and Orthometric Heights over

Evaluation of EGM08 based on GPS and orthometric heights
over the Hellenic mainland (*)
C. Kotsakis and K. Katsambalos
Department of Geodesy and Surveying
School of Engineering
Aristotle University of Thessaloniki
Univ. Box 440, Thessaloniki 541 24, Greece
M. Gianniou
Ktimatologio S.A. (Hellenic Cadastre)
339 Mesogion Ave.
Athens 152 31, Greece
ABSTRACT
This report presents the evaluation results for the new Earth Gravitational Model (EGM08)
that was recently released by the US National Geospatial-Intelligence Agency, using GPS
and leveled orthometric heights in the area of Greece. Detailed comparisons of geoid undulations obtained from the EGM08 model and other combined global geopotential models
(GGMs) with GPS/leveling data have been performed in both absolute and relative sense.
The test network covers the entire part of the Hellenic mainland and it consists of more than
1500 benchmarks which belong to the Hellenic national triangulation network, with direct
leveling ties to the Hellenic vertical reference frame. The spatial positions of these benchmarks have been recently determined at cm-level accuracy (with respect to ITRF2000) during a nation-wide GPS campaign that was organized in the frame of the HEPOS project.
Our results reveal that EGM08 offers a major improvement (more than 60%) for the
agreement among geoidal, ellipsoidal and orthometric heights over the mainland part of
Greece, compared to the performance of other combined GGMs for the same area.
(*) Final report submitted to the IAG/IGFS Joint Working Group “Evaluation of Global Earth Gravity Models”
(November 2008)
1
1. INTRODUCTION
The development of the Earth Gravitational Model EGM08 by the US National GeospatialIntelligence Agency (Pavlis et al. 2008) unveiled a major achievement in global gravity
field mapping. For the first time in modern geodetic history, a spherical harmonic model
complete to degree and order 2159, with additional spherical harmonic coefficients (SHCs)
extending up to degree 2190 and order 2159, is available for the representation of the
Earth’s external gravitational potential. This new model offers an unprecedented level of
spatial sampling resolution (~ 9 km) for the recovery of gravity field functionals over the
entire globe, and it contributes in a most successful way to the continuing efforts of the geodetic community for a high-resolution and high-accuracy reference model of Earth’s mean
gravity field.
Following the official release of EGM08 to the Earth science community, there is a strong
interest among geodesists to quantify its actual accuracy with different validation techniques and ‘external’ data sets, independently of the estimation and error calibration procedures that were used for its development. In response to the above interest and as part of the
related activities that have been coordinated by the IAG/IGFS Joint Working Group on the
Evaluation of Global Earth Gravity Models, the objective of this report is to present the
EGM08 evaluation results that have been obtained for the area of Greece using GPS and
leveled orthometric heights. A brief summary of these results has already been given in
Kotsakis et al. (2008), lacking though a number of additional tests that are presented for the
first time herein (see Sect. 4).
The test network consists of 1542 control points that belong to the Hellenic national triangulation frame, with direct ties to the Hellenic national vertical reference frame through
spirit (and in some cases trigonometric) leveling surveys. These control points were recently re-surveyed through a national GPS campaign in the frame of the HEPOS project
(more details to be given in Sect. 2) and their spatial positions have been estimated anew at
cm-level accuracy with respect to ITRF2000.
Some key features of our study are the extensive national coverage and high spatial density
of the test network, corresponding approximately to an average distance of 7 km between
adjacent points throughout Greece (Figure 1). These characteristics have been most helpful
in identifying the significant improvement that EGM08 yields, over other existing geopotential models, for the representation of gravity field features in certain Hellenic mountainous areas (see Sect. 3). This is actually the first time that a detailed quality analysis for the
performance of global geopotential models (GGMs) is carried out over the entire Hellenic
mainland with the aid of precise GPS positioning. Consequently, our study also provides a
preliminary, yet reliable, assessment about the feasibility of EGM08 for determining orthometric height differences via GPS/geoid-based leveling techniques in Greece (see Sect.
5).
2. DATA SETS
All the evaluation tests and their corresponding results that are presented in the following
sections refer to a network of 1542 GPS/leveling benchmarks which covers the entire
mainland region of Greece with a relatively uniform spatial distribution (see Figure 1).
2
Note that some control points which were originally existing in this network, but they were
later identified as ‘problematic’ (mainly due to suspected blunders in their orthometric
heights that are provided by the Hellenic Military Geographic Service), have been removed
from the following analysis and they are not included in the test network shown in Figure 1.
Although a large number of additional GPS/leveling benchmarks were also available in the
Greek islands, they have been deliberately excluded from our current analysis to avoid misleading systematic effects in the evaluation results due to unknown vertical datum differences that exist between the various islands and the mainland region.
Figure 1. Geographical distribution of the 1542 GPS/leveling benchmarks
over the Hellenic mainland.
2.1 Ellipsoidal heights
Within the frame of currently ongoing efforts for the enhancement of the spatial data infrastructure in Greece, a national GPS campaign took place in 2007 in order to acquire a sufficient number of control points with accurately known 3D spatial positions in an ITRF-type
coordinate system. These activities have been initiated by the Ministry for the Environment,
Planning and Public Works and the financial support of the EU and the Hellenic State, and
they are part of the HEPOS (Hellenic Positioning System) project that will lead to the
launch of a modern satellite-based positioning service for cadastral, mapping, surveying
and other geodetic applications in Greece (Gianniou 2008). The entire project is coordinated by Ktimatologio S.A, a state-owned private sector firm that is responsible for the operation of the Hellenic Cadastral system.
The aforementioned GPS campaign involved more than 2450 geodetic benchmarks within
the existing national triangulation network, part of which are the 1542 points shown in Figure 1. The main scope of the campaign was to provide an ample number of control stations
3
for determining a precise datum transformation model between the official Hellenic Geodetic Reference Frame of 1987 and other ITRF/ETRF-type frames. The actual fieldwork
was performed within a 6-month period (March to September 2007) using twelve dualfrequency Trimble 5700/5800 GPS receivers with Zephyr or R8 internal antennas. Thirty
three points were used as ‘base’ reference stations with 24-hour continuous GPS observations, while the rest of the control points were treated as ‘rover’ stations with observation
periods ranging between 1-3 hours. In all cases, a 15-sec sampling rate and an 15° elevation
cut-off angle were used for the data collection. Note that the maximum GPS-baseline length
that was obtained from the above procedure did not exceed 35 km.
After the processing of the GPS observations using EUREF/EPN ties and IGS precise orbits, the geocentric Cartesian coordinates of all stations (including the 1542 points shown in
Figure 1) were determined in ITRF2000 (epoch: 2007.236) and their geometric heights
were subsequently derived with respect to the GRS80 ellipsoid. The accuracy of the ellipsoidal heights ranges between 2-5 cm, while the horizontal positioning accuracy with respect to ITRF2000/GRS80 is marginally better by 1-2 cm (1σ level).
2.2 Orthometric heights
Helmert-type orthometric heights at the 1542 test points have been determined through leveling ties to surrounding benchmarks of the national vertical reference frame. These local
survey ties were performed in previous years by the Hellenic Military Geographic Service
(HMGS) using spirit and/or trigonometric leveling techniques. It should be mentioned that
a large number of the test points is located in highly mountainous areas (i.e. 24% of them
have orthometric heights H > 800 m).
The quality of the known orthometric heights in our test network is mainly affected by two
factors: the internal accuracy and consistency of the Hellenic vertical datum (HVD), and
the observation accuracy of the local leveling ties to the surrounding HVD benchmarks.
Due to the absence of sufficient public documentation from the part of HMGS, the absolute
accuracy of these orthometric heights is largely unknown. Their values refer, in principle,
to the equipotential surface of Earth’s gravity field that coincides with the mean sea level at
the HVD’s fundamental tide-gauge reference station located in Piraeus port (unknown Wo
value, period of tide gauge measurements: 1933-1978); for more details, see Antonopoulos
et al. (2001), Takos (1989).
2.3 GPS-based geoid undulations
Based on the known ellipsoidal and orthometric heights, geoid undulations have been computed at the 1542 test points according to the equation
N GPS = h − H
(1)
The above values provide the ‘external’ dataset upon which the following EGM08 validation tests will be performed.
Note that low-pass filtering or other smoothing techniques have not been applied to the
GPS/H geoid heights (NGPS). As a result, the effect of the omission error associated with all
tested GGMs will be directly reflected in our evaluation results.
4
2.4 GGM-based geoid undulations
Geoid undulations have also been computed at the 1542 GPS/leveling benchmarks using
several different GGMs. For the evaluation results presented herein, we consider the most
recent ‘mixed’ GGMs that have been produced from the combined analysis of various types
of satellite data (CHAMP, GRACE, SLR), terrestrial gravity data, and altimetry data; see
Table 1.
Table 1. GGMs used for the tests at the 1542 Hellenic GPS/leveling benchmarks.
Models
EGM08
EIGEN-GL04C
EIGEN-CG03C
EIGEN-CG01C
GGM02C
EGM96
nmax
2190
360
360
360
200
360
Reference
Pavlis et al. (2008)
Förste et al. (2006)
Förste et al. (2005)
Reigber et al. (2006)
Tapley et al. (2005)
Lemoine et al. (1998)
The determination of GGM geoid undulations was carried out through the general formula
(Rapp 1997)
N =ζ +
∆g FA − 0.1119 H
H + No
γ
(2)
where ζ and ∆gFA denote the height anomaly and free-air gravity anomaly signals, which
are computed from spherical harmonic series expansions (up to nmax) based on the SHCs of
each model and the GRS80 normal gravity field parameters. Only the gravitational potential
coefficients with degrees n≥2 were considered for these harmonic synthesis computations,
excluding the contribution of the zero/first-degree harmonics from the GGM-based signals.
Note that EIGEN-CG01C and EIGEN-CG03C are the only models among the tested GGMs
which are accompanied by non-zero first-degree SHCs. Nevertheless, their omission in the
computation of the ζ values has a negligible effect (mm-level) in our evaluation tests.
The term No represents the contribution of the zero-degree harmonic to the GGM geoid undulations with respect to the GRS80 reference ellipsoid. It is computed according to the
well known formula (e.g. Heiskanen and Moritz 1967)
No =
GM − GM o Wo −U o
−
Rγ
γ
(3)
where the parameters GMo and Uo correspond to the Somigliana-Pizzeti normal gravity
field generated by the GRS80 ellipsoid (Moritz 1992)
GMo = 398600.5000 × 109 m3 s-2
Uo = 62636860.85 m2 s-2
The Earth’s geocentric gravitational constant (GM) and the constant gravity potential of the
geoid (Wo) have been set to the following values
5
GM = 398600.4415 × 109 m3 s-2
Wo = 62636856.00 m2 s-2
(IERS Conventions 2003)
while the mean Earth radius R and the mean normal gravity γ on the reference ellipsoid are
taken equal to 6371008.771 m and 9.798 m s-2, respectively (GRS80 values). Based on the
above conventional choices, the zero-degree term from Eq. (3) yields the value No = -0.442
m, which has been added to the geoid undulations obtained from the corresponding SHC
series expansions of all GGMs.
Remark. The numerical computations for the spherical harmonic synthesis of the N values
from the various GGMs have been performed with the ‘harmonic_synth_v02’ software program that is freely provided by the EGM08 development team (Holmes and Pavlis 2006).
Note also that the final GGM geoid undulations obtained from Eq. (2) refer to the zero-tide
system, with respect to a geometrically fixed reference ellipsoid (GRS80).
2.5 Height data statistics
The statistics of the individual height datasets that will be used in our evaluation tests are
given in Table 2. Note that the statistics for the GGM geoid undulations refer to the values
computed from Eq. (2) at the 1542 GPS/leveling benchmarks using the full spectral resolution of each model.
From the following table (see, in particular, the mean values in the fourth column) it is evident the existence of a large discrepancy (> 25 cm) between the reference surface of the
Hellenic vertical datum (which is associated with an unknown Wo value) and the equipotential surface of Earth’s gravity field that is specified by the IERS conventional value Wo =
62636856.00 m2 s-2 and realized by the various GGMs over the Hellenic mainland region.
Table 2. Statistics of various height datasets over the test network of 1542 Hellenic GPS/leveling
benchmarks (units in m).
h
H
NGPS = h-H
N (EGM08)
N (EIGEN-GL04C)
N (EIGEN-CG03C)
N (EIGEN-CG01C)
N (GGM02C)
N (EGM96)
Max
2562.753
2518.889
43.864
44.374
44.104
44.049
44.108
44.034
44.007
Min
24.950
0.088
19.481
19.663
19.303
19.257
19.663
19.771
19.687
Mean
545.676
510.084
35.592
35.968
35.874
35.861
35.823
35.905
36.037
σ
442.418
442.077
5.758
5.800
5.878
5.867
5.873
5.780
5.753
It is also interesting to observe the considerable mean offset of the full-resolution EGM08
geoid (nmax = 2190) with respect to the geoid realizations obtained from other GGMs at the
GPS/leveling benchmarks. This offset varies from 6 to 15 cm and it should be attributed to
long/medium-wavelength systematic differences between EGM08 and the other GGMs
over the Hellenic area.
6
3. POINTWISE EVALUATION TESTS
AFTER A SIMPLE BIAS FIT
A series of GGM evaluation tests were performed based on the point values for the ellipsoidal and orthometric heights in the control network. The statistics of the differences between the GPS-based and the GGM-based geoid heights are given in Table 3. In all cases,
the values shown in this table refer to the statistics after a least-squares constant bias fit was
applied to the original misclosures h-H-N at the 1542 Hellenic GPS/leveling benchmarks.
The differences in the estimated bias obtained from each model (see last column in Table 3)
indicate the existence of systematic regional offsets among the GGM geoids that are likely
caused by long/medium-wavelength commission errors in their SHCs and additional omission errors due to their limited spectral resolution. Furthermore, the actual magnitude of the
bias between NGPS and N suggests the presence of a sizeable offset between (a) the equipotential surface associated with the IERS conventional value Wo = 62636856.00 m2s-2 and
realized by the various GGMs over the Hellenic region, and (b) the HVD reference surface
that is realized through the GPS/H geoid heights NGPS at the test points. For example, based
on the results from the full-resolution version of the new model, the HVD reference surface
appears to be located 38 cm below the EGM08/Wo/GRS80 geoid realization.
Table 3. Statistics of the residuals NGPS−N (after a least-squares constant bias fit)
at the 1542 GPS/leveling benchmarks (units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.542
1.476
1.773
1.484
1.571
2.112
1.577
Min
-0.437
-1.287
-1.174
-1.173
-1.135
-1.472
-1.063
σ
0.142
0.370
0.453
0.453
0.492
0.551
0.423
Bias
-0.377
-0.334
-0.283
-0.270
-0.231
-0.313
-0.446
From the results given in the above table, it is evident that EGM08 offers a remarkable improvement for the agreement among ellipsoidal, orthometric and geoidal heights in Greece.
Compared to other GGMs, the standard deviation of the EGM08 residuals NGPS-N over the
test network decreases by a factor of 3 (or more). The improvement obtained from the new
model is visible even in its 30' limited-resolution version (nmax=360), which matches the
GPS/H geoid within ±37 cm (in an average pointwise sense), while all previous GGMs of
similar resolution do not perform better than ±42 cm. The major contribution, however,
comes from the ultra-high frequency band of EGM08 (360 < n < 2190) which enhances the
consistency between GGM and GPS/H geoid heights at ±14 cm (1σ level).
In Table 4, we can see the percentage of the GPS/leveling benchmarks whose adjusted residuals h-H-N (after a constant bias fit) fall within a specified range of geoid uncertainty.
The agreement between EGM08 and GPS/H geoid heights is better than 10 cm for more
than half of the total 1542 test points, whereas for the other GGMs the same consistency
level is only reached at 18% (or less) of the test points. Furthermore, almost 85% of the test
points give an agreement between the full-resolution EGM08 geoid and the GPS/leveling
7
data that is better than 20 cm, compared to 36% (or less) in the case of all other global
models that were tested.
Table 4. Percentage of the 1542 test points whose absolute values of their adjusted residuals
NGPS−N (after a least-squares constant bias fit) are smaller than some typical
geoid accuracy levels.
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
< 2 cm
13.3 %
4.5 %
3.6 %
3.3 %
2.9 %
2.9 %
4.3 %
< 5 cm
29.8 %
11.6 %
9.3 %
8.3 %
7.8 %
7.4 %
9.8 %
< 10 cm
53.5 %
22.8 %
17.7 %
17.5 %
15.1 %
15.0 %
17.5 %
< 15 cm
73.0 %
32.7 %
27.5 %
26.5 %
23.0 %
22.6 %
27.7 %
< 20 cm
84.6 %
43.7 %
36.0 %
34.7 %
29.4 %
30.2 %
35.5 %
The horizontal spatial variations of the (full-resolution) EGM08 residuals NGPS-N did not
reveal any particular systematic pattern within the test network. Both their latitudedependent and longitude-dependent scatter plots, as shown in Figures 2 and 3, are free of
any sizeable north/south or east/west tilts over the Hellenic mainland. In other GGMs, however, some strong localized tilts and systematic oscillations can be identified in the NGPS-N
residuals, mainly due to larger commission errors associated with their SHCs and significant omission errors involved in the recovery of the geoid signal (see Figures 2 and 3).
Our evaluation results have also confirmed that EGM08 performs exceedingly better than
the other models over the mountainous parts of the Hellenic test network. A strong indication can be seen in the scatter plots of the pointwise residuals NGPS-N (after the constant bias
fit) with respect to the orthometric heights of the corresponding GPS/leveling benchmarks
(Figure 4). These plots reveal a height-dependent bias between the GGM and GPS/H geoid
heights, which is considerably reduced in the case of EGM08. Apparently, the higher frequency content of the new model gives a better approximation for the terrain-dependent
gravity field features over Greece, a fact that is visible from the comparative analysis of the
scatter plots in Figure 4. The remaining height-dependent linear trend in the full-resolution
EGM08 residuals NGPS-N (see Figure 4) is caused not only by commission/omission model
errors, but it reflects also existing systematic problems in the orthometric heights of the
tests points.
Further manifestation for the correlation of GGM and GPS/H geoid differences with the
topographic height of the test points can be found in the color plots given in Figure 5. With
the visual aid of the ETOPO2 digital elevation model, it is seen that larger values for the
residuals NGPS-N occur mostly over geographical areas with strong topographic features.
Note that the spatial distribution of the geoid height residuals for the full-resolution model
EGM08 is depicted in two separate plots, each with a different color-scaling scheme. From
the first of these plots, we can verify the overall improvement in the geoid representation
over the Hellenic mountains that is achieved with EGM08, compared to the performance of
previous GGMs over the same areas. The second scatter plot of the EGM08 geoid residuals
NGPS-N (see lower left corner in Figure 5) reveals the remaining inconsistencies with the
GPS/leveling data, which are caused by the commission/omission errors of the new model
and other unknown systematic distortions in the orthometric heights at the test points.
8
EGM08
(nmax=2190)
EGM08
(nmax=360)
EGM96
(nmax=360)
EIGEN-CG03C
(nmax=360)
GGM02C
(nmax=200)
EIGEN-GL04C
(nmax=360)
Figure 2. Latitude-dependent variations of the residuals NGPS−N
(after a least-squares constant bias fit) at the 1542 GPS/leveling benchmarks.
9
EGM08
(nmax=2190)
EGM08
(nmax=360)
EGM96
(nmax=360)
EIGEN-CG03C
(nmax=360)
GGM02C
(nmax=200)
EIGEN-GL04C
(nmax=360)
Figure 3. Longitude-dependent variations of the residuals NGPS−N
(after a least-squares constant bias fit) at the 1542 GPS/leveling benchmarks.
10
EGM08
(nmax=2190)
EGM08
(nmax=360)
EGM96
(nmax=360)
EIGEN-CG03C
(nmax=360)
GGM02C
(nmax=200)
EIGEN-GL04C
(nmax=360)
Figure 4. Height-dependent variations of the residuals NGPS−N
(after a least-squares constant bias fit) at the 1542 GPS/leveling benchmarks.
11
EGM08
(nmax=2190)
EGM08
(nmax=360)
EGM96
(nmax=360)
EIGEN-GL04C
(nmax=360)
EGM08
(nmax=2190)
ETOPO2 DTM
(*) note the different color scaling
compared to the plots shown above
Figure 5. Colored scatter plots showing the geographical distribution of the differences NGPS−N
(after a least-squares constant bias fit) at the 1542 GPS/leveling benchmarks.
12
4. POINTWISE EVALUATION TESTS
WITH DIFFERENT PARAMETRIC MODELS
In addition to the evaluation results that were presented in the previous section, another set
of numerical experiments has been carried out using a number of different parametric models for the least-squares adjustment of the differences NGPS – N. The motivation for these
additional tests was to investigate the fitting performance of some known linear models that
are frequently used in geoid evaluation studies with heterogeneous height data, and to assess their feasibility in modeling the systematic discrepancies between the GGM and
GPS/H geoid surfaces over the Hellenic mainland. Although these tests were implemented
with all six GGMs that were initially selected for our study, only the results obtained with
EGM08 and EGM96 will be presented herein due to space limitations.
The various parametric models that have been fitted to the original misclosures h-H-N are
given in Eqs. (5)-(10). Model 1 uses a single constant-bias parametric term and it is actually
the same model that was employed for all tests of the previous section. Model 2 incorporates two additional parametric terms which correspond to an average north-south and eastwest tilt between the GGM and GPS/H geoids. Model 3 is the usual ‘4-parameter model’
which geometrically corresponds to a 3D spatial shift and an approximate uniform scale
change of the GGM’s reference frame with respect to the underlying reference frame of the
GPS heights (or vice versa). Finally, models 4, 5 and 6 represent height-dependent linear
corrector surfaces that constrain the relation among ellipsoidal, orthometric and geoidal
heights in terms of the generalized equation
h – (1+δsH)H – (1+δsN)N = µ
(4)
The above equation takes into consideration the fact that the spatial scale of the GPS
heights does not necessarily conform with the spatial scale induced by the GGM geoid undulations and/or the inherent scale of the orthometric heights obtained from terrestrial leveling techniques. Moreover, the GGM geoid undulations and/or the local orthometric heights
are often affected by errors that are correlated, to a certain degree, with the Earth’s topography (see the results in Figures 4 and 5), a fact that can additionally justify the use of model
4 or 6 for the optimal fitting between NGPS and N.
Model 1
hi − H i − N i = µ + vi
(5)
Model 2
hi − H i − N i = µ + a (ϕ i − ϕ o ) + b(λi − λ o ) cos ϕ i + vi
(6)
Model 3
hi − H i − N i = µ + a cos ϕ i cos λi + b cos ϕ i sin λi + c sin ϕ i + vi
(7)
Model 4
hi − H i − N i = µ + δsH H i + vi
(8)
Model 5
hi − H i − N i = µ + δs N Ni + vi
(9)
13
Model 6
(10)
hi − H i − N i = µ + δsH H i + δsN Ni + vi
Remark. A combination of the above models (e.g. the ‘4-parameter’ or the ‘bias and tilt’
model merged with a height-dependent scaling term) may also be useful in practice, depending on the behavior of the actual data.
The statistics of the adjusted residuals {vi} in the test network of 1542 Hellenic
GPS/leveling benchmarks, after the least-squares fitting of the previous parametric models,
are given in Tables 5 and 6 for the case of EGM96 and EGM08, respectively.
Table 5. Statistics of the differences NGPS−N for the EGM96 geoid heights, after the
least-squares fitting of various parametric models at the 1542 GPS/leveling
benchmarks (units in m).
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Max
1.577
1.587
1.681
1.198
1.572
1.176
Min
-1.063
-1.073
-1.097
-0.847
-1.053
-0.861
σ
0.423
0.422
0.411
0.341
0.423
0.341
Bias (µ)
-0.446
-0.445
303.983
-0.735
-0.381
-0.656
Table 6. Statistics of the differences NGPS−N for the EGM08 geoid heights, after the
least-squares fitting of various parametric models at the 1542 GPS/leveling
benchmarks (units in m).
Model 1
Model 2
Model 3
Model 4
Model 5
Model 6
Max
0.542
0.521
0.522
0.480
0.528
0.474
Min
-0.437
-0.398
-0.398
-0.476
-0.442
-0.421
σ
0.142
0.137
0.137
0.131
0.135
0.123
Bias (µ)
-0.377
-0.377
3.479
-0.440
-0.109
-0.160
From the above results, it can be concluded that the low-order parametric models which are
commonly used in the combined adjustment of GPS, geoid and leveled height data (models
2 and 3) do not offer any significant improvement for the overall fitting between the
EGM08 geoid (or the EGM96 geoid) and the GPS/leveling heights over the Hellenic
mainland. On the other hand, a purely height-dependent parametric model (model 6) enhances the statistical fit between the EGM08 and the EGM96 geoid with the GPS/leveling
heights by 2 cm and 8 cm, respectively (i.e. compared to the performance of the bias-only
model 1). The improvement in the sigma values obtained from models 4 and 6 should be
attributed to the elimination of the linear correlation trend that was previously identified
(see Figure 4) between the misclosures h-H-N and the orthometric heights of the test points.
14
Note that all alternative models which are tested in this section include a common parametric term in the form of a single constant bias. However, the various estimates of the common bias parameter µ, as obtained from the least-squares adjustment of each model, exhibit
significant variations among each other (see last column in Tables 5 and 6). Specifically,
the estimated bias between NGPS and N which is computed from the usual ‘4-parameter’
model appears to be highly inconsistent with respect to the corresponding estimates from
the other parametric models. This is not surprising since the intrinsic role of the bias µ in
model 3 is not to represent the average spatial offset between the GGM and the GPS/H geoids, as it happens for example in the case of model 1. In fact, the three additional parametric terms in model 3 are the ones that absorb the systematic part of the differences NGPS −N
in the form of a three-dimensional spatial shift (a → tx, b → ty, c → tz), leaving to the fourth
bias parameter µ the role of a ‘scale-change’ effect.
At this point, it is perhaps instructive to recall the linearized transformation formula for geoid heights between two parallel geodetic reference frames (see, e.g., Kotsakis 2008)
N i′ − N i = (awi + N i )δs + t x cos ϕ i cos λi + t y cos ϕ i sin λi + t z sin ϕ i
(11)
where a denotes the semi-major axis of the common reference ellipsoid, δs is the differential scale change between the underlying frames, and wi corresponds to the auxiliary
unitless term (1 − e 2 sin 2 ϕ i )1/ 2 that is approximately equal to 1 (i.e. the squared eccentricity
of the reference ellipsoid is e2 ≈ 0.0067). The above formula conveys, in the language of
geodetic datum transformation, the basic geometric principles of the ‘4-parameter’ model
that is frequently employed for the optimal fitting of GPS, geoid and leveled height data.
Given the analytic expression in Eq. (11), the constant bias µ that appears in the formulation
of model 3 emulates the effect of a mean spatial re-scaling rather than a mean spatial offset
between two different geoid realizations.
Although less inconsistent with each other, the estimates of the bias parameter µ from the
other parametric models show dm-level fluctuations in their values. It should be noted
though that the inclusion of additional spatial tilts for the fitting between NGPS and N does
not distort the initial estimate of µ that was obtained from model 1 over the Hellenic
mainland. On the other hand, the use of height-dependent scaling terms (models 4, 5 and 6)
affects considerably the final estimates of the bias parameter µ, as it can be easily verified
from the results in Tables 5 and 6.
All in all, the problem of obtaining a realistic estimate for the average spatial offset between a local vertical datum (e.g. HVD in our case) and a GGM geoid seems to have a
strong dependence on the parametric model that is used for the adjustment of heterogeneous
height data over a test network of GPS/leveling benchmarks. Since there exist strong theoretical and practical arguments that can be stated in favor of the generalized constraint in
Eq. (4), the use of the simple model 1 is not necessarily the safest choice for estimating the
average spatial offset between GGM and GPS/H geoids over a regional network. In view of
the frequent absence (or even ignorance) of a complete and reliable stochastic error model
for the properly weighted adjustment of the differences NGPS −N, a clear geometrical interpretation of the estimated bias µ is not always a straightforward task in GGM evaluation
studies.
15
5. BASELINE EVALUATION TESTS
An additional set of evaluation tests was also performed through the comparison of GGM
and GPS/H geoid slopes over the Hellenic network of 1542 GPS/leveling benchmarks. For
all baselines formed within this network, the following differences of relative geoid undulations were determined
∆N ijGPS − ∆N ij = (h j − H j − hi + H i ) − ( N j − N i )
(12)
Note that the computation of the above differences took place after the implementation of a
least-squares bias/tilt fit between the pointwise values of the GGM and GPS/H geoid
heights.
Depending on the actual baseline length, the residual values from Eq. (12) were grouped
into various spherical-distance classes and their statistics were then evaluated within each
class. Given the actual coverage and spatial density of the GPS/leveling benchmarks in our
test network, baselines with length from 2 km up to 600 km were considered for this
evaluation scheme. The statistics of the differences between the GGM and GPS/H relative
geoid heights, for five selected baseline classes, are given in the following tables.
Table 7. Statistics of the differences between GGM and GPS/H relative geoid heights
for baselines with length < 3 km (number of baselines: 47, units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.142
0.140
0.156
0.148
0.152
0.137
0.136
Min
-0.156
-0.206
-0.200
-0.205
-0.207
-0.230
-0.199
σ
0.058
0.080
0.087
0.087
0.087
0.081
0.081
Bias
-0.009
-0.018
-0.015
-0.016
-0.016
-0.021
-0.014
Table 8. Statistics of the differences between GGM and GPS/H relative geoid heights
for baselines with length < 5 km (number of baselines: 289, units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.643
0.648
0.649
0.643
0.640
0.685
0.643
Min
-0.474
-0.534
-0.542
-0.540
-0.536
-0.571
-0.553
σ
0.111
0.154
0.155
0.155
0.156
0.162
0.154
Bias
0.006
0.003
0.005
0.005
0.005
0.003
0.005
16
Table 9. Statistics of the differences between GGM and GPS/H relative geoid heights
for baselines with length 5-10 km (number of baselines: 2119, units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.465
1.022
0.983
0.971
0.976
0.967
0.963
Min
-0.629
-1.044
-0.988
-1.026
-1.039
-0.991
-1.002
σ
0.125
0.248
0.251
0.251
0.252
0.264
0.251
Bias
0.001
-0.004
-0.000
-0.001
-0.002
0.002
0.003
Table 10. Statistics of the differences between GGM and GPS/H relative geoid heights
for baselines with length 10-50 km (number of baselines: 56575, units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.859
2.778
2.480
2.335
2.335
3.221
2.532
Min
-0.781
-2.417
-2.430
-2.488
-2.445
-2.760
-2.393
σ
0.164
0.514
0.552
0.550
0.555
0.627
0.542
Bias
-0.001
-0.012
-0.019
-0.021
-0.021
-0.012
-0.013
Table 11. Statistics of the differences between GGM and GPS/H relative geoid heights
for baselines with length 50-100 km (number of baselines: 135970, units in m).
EGM08 (nmax=2190)
EGM08 (nmax=360)
EIGEN-GL04C (nmax=360)
EIGEN-CG03C (nmax=360)
EIGEN-CG01C (nmax=360)
GGM02C (nmax=200)
EGM96 (nmax=360)
Max
0.891
2.332
2.410
2.172
2.356
3.043
2.226
Min
-0.881
-2.282
-2.773
-2.568
-2.600
-3.611
-2.480
σ
0.189
0.552
0.658
0.651
0.668
0.834
0.623
Bias
-0.003
-0.013
-0.041
-0.043
-0.037
-0.067
-0.028
As seen from the results in Tables 7 through 11, the full-resolution EGM08 model performs
consistently better than all other GGMs over all baseline classes. The improvement becomes more pronounced as the baseline length increases, indicating the significant contribution of the EGM08 high-degree harmonics (n > 360) for the slope representation of the
Hellenic geoid over baselines 5-100 km. For example, the resultant σ values of the differences ∆NGPS −∆N are reduced by a factor of 1.4 for baselines <5 km, by a factor of 2 for
baselines 5-10 km, and by a factor of about 3.5 for baselines 10-100 km (compared to the
performance of EGM96 and other EIGEN-type models).
It is also interesting to observe the considerable bias in the geoid slope residuals ∆NGPS −∆N
obtained from all tested GGMs (except from the full-resolution EGM08 model) for base-
17
lines 10-100 km. This result should be attributed to existing systematic errors in the medium-wavelength SHCs of the tested GGMs and additional omission errors in the preEGM08 models, which produce an apparent scale difference between GGM and GPS/H
relative geoid undulations for the aforementioned baseline range.
The overall behaviour of the sigma values for the differences between GGM and GPS/H
geoid slopes is shown in Figure 6, over all baseline classes that were considered in our
tests. The remarkable improvement in the relative geoid accuracy from the EGM08 model
is clearly visible, indicating an ∆N-consistency level with the external GPS/leveling data
that varies from ±6 cm to ±20 cm (1σ level).
0.9
EGM08
0.8
EIGEN-GL04C
0.7
EIGEN-CG03C
(m
)
(m)
0.6
EIGEN-CG01C
0.5
GGM02C
0.4
EGM96
0.3
0.2
0.1
0
1
10
100
1000
Baseline
length
(in km,
log-scale)
Baseline
Length
(km)
Figure 6. Std of the differences ∆N ijGPS − ∆N ij in the test network
of 1542 GPS/leveling benchmarks, as a function of the baseline length.
Focusing on the geoid-slope evaluation results for short baselines (up to 30 km) can give us
an indication for the expected accuracy in GPS/leveling projects when using an EGM08
reference geoid model over Greece. Our preliminary analysis in the test network showed
that the agreement between the height differences ∆Hij computed: (a) directly from the
known orthometric heights at the GPS/levelling benchmarks and (b) indirectly from the
GPS/EGM08 ellipsoidal and geoid heights, could be approximated by the statistical error
model σ∆H = σo L1/2 with the a-priori sigma factor σo ranging between 3-5 cm/km (for baseline length L<30 km). Although such a performance cannot satisfy mm-level accuracy requirements for vertical positioning (which are ‘easily’ achievable through spirit leveling
techniques), it nevertheless provides a major step forward that can successfully accommodate a variety of engineering and surveying applications. Note that the corresponding performance of EGM96 in our test network is described by a relative accuracy factor of σo ≈ 9
cm/km for baselines <30 km.
18
6. Conclusions
The results of our evaluation tests have revealed the superiority of EGM08 over all existing
mixed GGMs for the area of Greece. The new model outperforms the other tested GGMs at
the 1542 Hellenic GPS/leveling benchmarks and it improves the statistical fit with the Hellenic GPS/H geoid by approximately 30 cm (or more)! The pointwise agreement among
ellipsoidal, orthometric and EGM08-based geoid heights is at ±14 cm (1σ level), reflecting
mainly the regional effects of the commission errors in the model’s SHCs, as well as other
local distortions in the HVD orthometric heights at the control points.
In terms of relative geoid accuracy, EGM08 shows a rather stable performance for the standard deviation of the slope residuals ∆NGPS −∆N over all baseline lengths that were considered in our study. Compared to other tested GGMs whose relative geoid accuracy decreases
continuously over baselines 5-100 km (estimated values for σ∆N reach up to 60 cm), the
full-resolution EGM08 model gives a more balanced behavior with the corresponding values of σ∆N not exceeding 20 cm, even for baselines up to 600 km.
In conclusion, the results presented herein provide a promising testament for the future use
of EGM08 in geodetic applications over the Hellenic mainland. However, in view of its
possible forthcoming implementation in GPS-based leveling projects throughout Greece (in
conjunction with the HEPOS system), a more detailed analysis with additional interpolation
methods and spatial ‘corrector surfaces’ for modeling the differences NGPS −N or ∆NGPS −∆N
is required to achieve cm-level consistency for the transformation between GPS/EGM08
and HVD orthometric heights.
Acknowledgements. The GPS and leveling data used for this study were kindly provided by
Ktimatologio S.A under an ongoing research collaboration with the Department of Geodesy
and Surveying, Aristotle University of Thessaloniki, in the frame of the HEPOS project.
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